Related papers: Huygens' equations and the gradient-flow equations…
We have studied the gradient-flow equations in information geometry from a point-particle perspective. Based on the motion of a null (or light-like) particle in a curved space, we have rederived the Hamiltonians which describe the…
The gradient-flow equations with respect to the potential functions in information geometry are reconsidered from the perspective of the Weyl integrable geometry. The pre-geodesic equations associated with the gradient-flow equations are…
We analyze gradient flows with jumps generated by a finite set of complete vector fields in involution using some Radon measures $u\in \mathcal{U}_a$ as admissible perturbations. Both the evolution of a bounded gradient flow $\{x^u(t,\l)\in…
Assuming a-priori a smooth generating vector field, we introduce a generally covariant measure of the flow geometry called the referential gradient of the flow. The main result is the explicit relation between the referential gradient and…
The gradient flow is the evolution of fields and physical quantities along a dimensionful parameter~$t$, the flow time. We give a simple argument that relates this gradient flow and the Wilsonian renormalization group (RG) flow. We then…
The gradient-flow dynamics of an arbitrary geometric quantity is derived using a generalization of Darcy's Law. We consider flows in both Lagrangian and Eulerian formulations. The Lagrangian formulation includes a dissipative modification…
Uncertainty propagation and filtering can be interpreted as gradient flows with respect to suitable metrics in the infinite dimensional manifold of probability density functions. Such a viewpoint has been put forth in recent literature, and…
We consider a general formulation of gradient flow evolution for problems whose natural framework is the one of metric spaces. The applications we deal with are concerned with the evolution of {\it capacitary measures} with respect to the…
This article serves to concisely review the link between gradient flow systems on hypergraphs and information geometry which has been established within the last five years. Gradient flow systems describe a wealth of physical phenomena and…
We analyse second order (in Riemann curvature) geometric flows (un-normalised) on locally homogeneous three manifolds and look for specific features through the solutions (analytic whereever possible, otherwise numerical) of the evolution…
We study linear evolution equations in separable Hilbert spaces defined by a bounded linear operator. We answer the question which of these equations can be written as a gradient flow, namely those for which the operator is real…
Motivated by recent developments in the fields of large deviations for interacting particle system and mean field control, we establish a comparison principle for the Hamilton--Jacobi equation corresponding to linearly controlled gradient…
Motivated by a constrained minimization problem, it is studied the gradient flows with respect to Hessian Riemannian metrics induced by convex functions of Legendre type. The first result characterizes Hessian Riemannian structures on…
We study the relationship between the classical Hamilton flow and the quantum Schr\"odinger evolution where the Hamiltonian is a degree-2 complex-valued polynomial. When the flow obeys a strict positivity condition equivalent to compactness…
This study leverages the basic insight that the gradient-flow equation associated with the relative Boltzmann entropy, in relation to a Gaussian reference measure within the Hellinger-Kantorovich (HK) geometry, preserves the class of…
Normal geodesic flows flows of Carnot-Caratheodory are discussed from the point of view of the theory of Hamiltonian systems. The geodesic flows corresponding to left-invariant metrics and left- and -right-invariant rank 2 distributions on…
The Wasserstein space of probability measures is known for its intricate Riemannian structure, which underpins the Wasserstein geometry and enables gradient flow algorithms. However, the Wasserstein geometry may not be suitable for certain…
Geometric flows have proved to be a powerful geometric analysis tool, perhaps most notably in the study of 3-manifold topology, the differentiable sphere theorem, Hermitian-Yang-Mills connections and canonical Kaehler metrics. In the…
A fundamental question in Riemannian geometry is to find canonical metrics on a given smooth manifold. In the 1980s, R. Hamilton proposed an approach to this question based on parabolic partial differential equations. The goal is to start…
The geometric evolution equations provide new ways to address a variety of non-linear problems in Riemannian geometry, and, at the same time, they enjoy numerous physical applications, most notably within the renormalization group analysis…